However, the compound eyes of shrimp and lobster show a tetragonal tile pattern, and the compound eye of the sea mantis is a mixture of tetragonal and hexagonal tile patterns. The compound eye of the fly also normally exhibits a hexagonal pattern, but in certain mutants it changes to a tetragonal pattern. These suggest that the tile pattern of the compound eye is not controlled solely according to physical stability. However, it is not known how the hexagonal and tetragonal patterns are controlled during development. .

The compound eyes of wild-type control flies show a hexagonal pattern, but in some mutants it changes to a tetragonal pattern. Since the tetragonal mutants have smaller compound eye, we hypothesized that the size of the compound eye itself is an essential factor that determines the tiling pattern.

Since the mutant compound eye is smaller along the vertical (dorsal-ventral) direction, we hypothesized that the small eye's connection with the head part stretches the eye tissue along the vertical direction, causing its elongation. In fact, we found that the vertical tension was enhanced in the tetragonal mutant eye, and the eye tissue was elongated along the verticl direction. However, the tension alone only causes the elongation of the hexagon, and does not explain the change from a hexagonal to a tetragonal pattern.

A geometrical partitioning method known as Voronoi diagram, which divides a region equally around multiple mother points on a plane, is used, for example, to determine school districts for elementary schools. The vertical bisectors of the line segments connecting the mother points (elementary schools) constitute the Voronoi edges, which divides the entire region (school district) equally forming the Voronoi diagram.

We have found that not only the control hexagonal tiles but also the mutant tetragonal tiles are accurately reproduced by Voronoi diagram. Although it is unreasonable to assume that the geometrical method of drawing vertical bisectors is performed in vivo, it is known that the exact same Voronoi diagram is drawn when each mother point grows concentrically and stops growing when the circles collide to form a boundary (previous figure). Combining experiments and computer simulations, we have demonstrated that the cells that make up the ommatidia expand like balloons, producing an effect similar to that of concentric growth.

Thus, arrangement of the ommatidia and their concentric growth determine the tiling pattern. When the ommatidia are evenly distributed along the vertical and horizontal axis, they form a hexagonal tile pattern. When they are stretched vertically, they form a tetragonal pattern. It was known that morphology of cell and tissue is regulated by gene function and physical constraints, but it is now clear that a geometrical mechanism plays an important role in addition to these factors, and that the tiling pattern of the compound eye is controlled by the coordinated action of these factors.